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Guest is certainly correct.

Another answer is 11_{base 26}

So

The smallest base 2 five digit palindrone is   10001₂     =   17₁₀

_{\(10001_2=17_{10}=33_8=11_{26}\)}

AB and CD are parallel. The length of a chord AB is 7 cm. AD = BC = 3 cm. DC is the diameter of a circle O. Find radius DO.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Let AF be the altitude of a trapezoid ABCD.

AO = DO => r          AB = 7       AD = BC = 3

FO = AB / 2  ==>    FO = 3.5         

AD² - (r - FO)² = r² - FO²        ==>        3² - (r - 3.5)² = r² - 3.5²       ==>  r = 4.5 cm

I "borrowed" Dragan's diagram and added a couple of red lines to make my answer even clearer. 

Happy Christmas

Ginger, you should post one of your memoirs.  The Christmas memoir you sent me a few years ago will fit the theme of your post. It too is delightfully funny, and it’s now a Christmas tradition for me to read it every year to my grandchildren.  I’d bet others would like to read it, and it might become a Christmas tradition for them too. 

I will forever and always enjoy reading:

Your narratives, particularly when they have artwork, and the artwork has particular pitchforks with errant math.

Let the midpoint of AB be M

Let angle COB = theta

Now look at right-angled triangle OMB.  

Use triangle MBO to fin an expression for cos(theta)

Use triangle BOC to find an expression for sin(theta)   (Use cosine rule)

No use sine^2(theta) + cos^2(theta) = 1

You should be able to finish it from there.   

Note that the answer is an approximation.  Maybe someone will come up with an exact answer.

That is excellent!

The number of possible paths is  10*3*3 = 90

You need to end on  0, 4, 8 or 12     no other multiples of  4 are possible

Say you choose the number 1   the only multiple of 4 you can get to is 0

    1 -1 +0    or     1+0-1    that is 2 favourable outcomes if you start with the number 1

Now do the same with all the other starting numbers to work out the number of favourable paths.

then the prob will be    number of favourable outcomes / 90

Please let guest finish this on their own

41  43  47  53  59  >>>Total= 5 primes between 40 and 60

41 + 20 =61 is prime

43 + 20=63 not prime

47 + 20 =67 is prime

53 + 20 =73 is prime

59 + 20 =79 is prime.

Probability = 4 / 5

Thank you, I learned something new today. 

Nicely done guest.  :)

By convention, the real part must go first.     -1+12i

What number must we add to 3-5i  to get 2+7i ? Here, i^2=-1

3+ -1=2

-5+ 12 = 7

so the answer is     -1+12i

For addition and subtraction you can treat the i like it is just an ordinary pronumeral.

We can turn this problem into a subtraction problem:  (2+7i) - (3-5i). 

Since the order of subtraction/addition problems don't matter (a+b=b+a and a-b=-b+a), we can turn the problem into: (2-3) + (7i - -5i)

Subtracting a negative number is like adding the positive version of it: (2-3) + (7i+5i)

Doing the parts individually, 2-3 = -1 and 7i+5i = 12i.

Finally, we can combine them and our solution is -1 + 12i or 12i - 1. 

I hope this helped :)

Nice try guest.

It took me quite a while to get my head around the question too.

The number is 3p

both 3 and p are prime numbers so  the only factors are  1,3, p and  3p

the sum of the factors is   4+4p

4+4p= 600

p+1=150

p=149    

600 - 3 = 597

597 x 1/3 =199 - which is your prime number p.

199 x 2 = 398

398 =[1, 2, 199, 398] - Number of its divisors.

Sum[1 + 2 + 199 + 398]== 600

A different interpretation.

10 to 99    is   90 numbers in total

which ones have BOTH digits either 8 or 9      

only 88, 89,98 and 99

so that is 4/90 

the wording is strange but this is how I have interpreted it.

11 001 010 100 101 011 +    Base 10

    110 100 011 000 100        Base 10

11 111 110 111 101 111           n

11 001 010 100 101 011 +    Base 2

    110 100 011 000 100        Base 2

11 111 110 111 101 111             m

n-m=0

This is just a computation problem: n = 11201201201221012 and m = 835718428, so n - m = 11201200365502584.

For the ellipse x^2/a^2 + y^2/b^2 = 1, a^2 + b^2 = c^2.  Using this formula, we get that the semi-major axis is 14.

 By Hamilton's Theorem, the solutions are z = 4^{1/4}*e^(pi*i/4), 4^{1/4}*e^(pi*i/4 + pi/4), 4^{1/4}*e^(pi*i/4 + 2*pi/4), and 4^{1/4}*e^(pi*i/4 + 3*pi/4).  Since 4^{1/4} = sqrt(2) and e^(pi*i/4) = (1 + i)/sqrt(2), the first solution is 1 + i.  Then the other roots work out as

4^{1/4}*e^(pi*i/4 + pi/4) = 1 - i,

4^{1/4}*e^(pi*i/4 + 2*pi/4) = -1 - i, and

4^{1/4}*e^(pi*i/4 + 3*pi/4) = -1 + i.

 By Hamilton's Theorem, the solutions are z = 4^{1/4}*e^(pi*i/4), 4^{1/4}*e^(pi*i/4 + pi/4), 4^{1/4}*e^(pi*i/4 + 2*pi/4), and 4^{1/4}*e^(pi*i/4 + 3*pi/4).  Since 4^{1/4} = sqrt(2) and e^(pi*i/4) = (1 + i)/sqrt(2), the first solution is 1 + i.  Then the other roots work out as

4^{1/4}*e^(pi*i/4 + pi/4) = 1 - i,

4^{1/4}*e^(pi*i/4 + 2*pi/4) = -1 - i, and

4^{1/4}*e^(pi*i/4 + 3*pi/4) = -1 + i.

The parallelogram has a base of 10 and a height of 18, so its area is 10*18 = 180.

By computer program

for (1 <= n <= 5000) {

  if n % 12 == 5 then a = a + 1

  if n % 13 == 1 then b = b + 1

  if n % 12 == 5 and n % 13 == 1 then c = c + 1

}

(a) The answer is a = 401.

(b) The answer is b = 374.

(c) The answer is c = 28.

To answer this question, you just need to subtract the number of words without vowels from the total number of words.

The total number of words is 6*6*6*6=1296

The total number of words without vowels is 4*4*4*4 = 256

Therefore, the answer to this question is 1296-256=1040 words.

The probability is 4/17.